introduction to perco lation

7/29/2019 Introduction to Perco Lation

http://slidepdf.com/reader/full/introduction-to-perco-lation 1/21

Introduction to

Percolation Theory

Danica Stojiljkovic



System in concern

•Discrete system in d dimensions

•Lattices

– 1D: array

– 2D: square, triangular, honeycomb

– 3D: cubic, bcc, fcc, diamond

– dD: hypercubic

– Bethe lattice (Cayley tree)



System in concern



System in concern

•

Each lattice site is occupiedrandomly and independently withprobability p

•Nearest neighbors: sites with oneside in common

•Cluster: a group of neighboringsites



Site and Bond Percolation

•

A “site” can be a field or a nodeof a lattice

•Bond percolation: bond ispresent with probability x

•Site-bond percolation:continuous transition betweensite and bond percolation



Percolation thresholds

• At some occupation probability pc a

spanning (infinite) cluster will appearfor the first time

• Percolation thresholds are different fordifferent systems

20 x 20 site lattice



Percolation thresholds



Cluster numbers

•

Number of cluster with s sites perlattice sites: ns

•For 1D lattice:

•In general:

•t – perimeter•gst –number of lattice animals with

size s and perimeter t

2

)1( p pn

s

s−=

∑ −=t

t s

st sp pgn )1(



Average cluster size

•

Probability that random sitebelongs to a cluster of size s is

w s=sn s / ∑ sn s

•

Average size of a cluster that wehit if we point to and arbitraryoccupied site that is not a part of an infinite cluster

S = ∑wss

= ∑ s2n s / ∑ sn s



Average cluster size

•

For 1D: S = (1+p)/(1-p)

~ | pc-p |-1

•For Bethe lattice:S ~ | pc-p |-1

•In generalS ~ | pc-p |-γ

•



Cluster numbers at pc

•

If ns at pc decays exponentially,than S would be finite, therefore:

ns(pc) ~ s-τ

•

We calculate for Bethens(p)/ ns(pc) ~ exp (-Cs)

whereC ~ | pc – p |1/ σ



Scaling assumption

•

Generalizing Bethe result:ns(p) ~ s-τ exp(-|p-pc|1/ σ s)

(1D result does not fit)

•We can derive all other criticalexponents from τ and σ

•Scaling assumptionns(p) ~ s-τ F(|p-pc|1/ σ s)

or ns(p) ~ s-τ ((p-pc)sΦ σ)



Universality

•

(z)Φ depends only ondimensionality, not on latticestructure

• τ and σ, as well as function (z)Φare universal

•Plotting ns(p)s-τ versus (p-pc)s σ would fall on the same line



Strength of the infinite network

•

P – probability of an arbitrary sitebelonging to the infinite network P + ∑ sns = p

•

Applicable only at p > pc•For Bethe lattice

P ~ (p – pc)

•In general

P ~ |p – pc|β•Spontaneous magnetization, difference betweenvapor and liquid density



Correlation function

•

g(r) -Probability that a site atdistance r from an occupied sitebelongs to the same cluster

•In 1D array:g(r) = pr = exp(-r/ ξ)

ξ = -1 / ln( p) = 1 /(pc-p)

• ξ - Correlation length



Radius of gyration

•

Average of the square radii

• r0 – center of mass•Relation of gyration radii andaverage site distance

∑=

−=

s

i

i

ss

r r R

1

2

02

∑−

=ij

ji

ss

r r R

2

2

22



Correlation length

•

Average distance between twosites belonging to the samecluster:

•

Near the percolation thresholdξ ~ | p – pc| -ν

∑∑∑

∑

=

=

r s

s ss

r

r

ns

ns R

r g

r gr

2

22

2

2

2

)(

)(

ξ



Fractal dimension

•

At p = pc : M ~ LD

where M is mass of the largestcluster in a box with sides L

• D<d and it is called fractaldimension

•For gyration radii:Rs ~ s1/D



Crossover phenomena

•

For finite lattices whose lineardimension L <<ξ , cluster behaves

as fractalsM ~ L D

•ξ acts like a measuring stick, andin its absence all the relevantfunctions becomes power laws

•For L >ξ,M ~ (L/ ξ)d ∙ ξ D ~ L d



References

•

Introduction to PercolationTheory, 2nd revised edition,1993by Dietrich Stauffer and AmnonAharony



To be continued…

•

Exact solution for 1D and Bethelattice

•Scaling assumption

•

Deriving relations betweencritical coefficients

•Numerical methods

•

Renormalization group

introduction to perco lation

Documents